Abstract : Let $D=G-K$ be a bounded symmetric domain and $S=K-L$ be its Shilov boundary We consider the action of $G$ with weight $ u\in\mathbb{Z}$ on functions on $D$ viewed as sections the line bundle and the corresponding eigenspace of $G$-invariant di erential operators. The Poisson transform maps hyperfunctions on the $S$ to the eigenspaces. We characterize the image in terms of Hua operators on the sections of the line bundle. For some special parameter the Poisson transform is of Szegö type mapping into the relative discrete series; we compute the corresponding elements in the discrete series.

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